Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility...

Rational Choice

CHOICE

1. Scarcity (income constraint)

2. Tastes (indifference map/utility function)

ECONOMIC RATIONALITY

The principal behavioral postulate is that a decision-maker chooses its most preferred alternative from those available to it.

The available choices constitute the choice set.

How is the most preferred bundle in the choice set located/found?

RATIONAL CONSTRAINED CHOICE

Affordablebundles

x1

x2

More preferredbundles


x1

x2

x1*

x2*


x1

x2

x1*

x2*

(x1*,x2*) is the mostpreferred affordablebundle.

E


MRS=x2/x1 = p1/p2

Slope of the indifference curve

Slope of the budget constraint

Individual’s willingness to trade

Society’s willingness to trade

At Equilibrium E


The most preferred affordable bundle is called the consumer’s ORDINARY DEMAND at the given prices and income.

Ordinary demands will be denoted byx1*(p1,p2,m) and x2*(p1,p2,m).


x1

x2

x1*

x2*

The slope of the indifference curve at (x1*,x2*) equals the slope of the budgetconstraint.


(x1*,x2*) satisfies two conditions: (i) the budget is exhausted, i.e.

p1x1* + p2x2* = m; and (ii) the slope of the budget constraint,

(-) p1/p2, and the slope of the indifference curve containing (x1*,x2*) are equal at (x1*,x2*).

COMPUTING DEMAND

How can this information be used to locate (x1*,x2*) for given p1, p2 and m?

Two ways to do this

1. Use Lagrange multiplier method

2. Find MRS and substitute into the Budget Constraint

COMPUTING DEMAND Lagrange Multiplier Method

Suppose that the consumer has Cobb-Douglas preferences

and a budget constraint given by

mxpxp 2211

aaxxxxU 12121 ),(


Aim

Set up the Lagrangian

mxpxpxx aa 2211

121 tosubject max

22111

21,,

21

xpxpmxxL aa

xx


Differentiate

(3) 0

(2) 0)1(

(1) 0

2211

2212

11

21

11

xpxpmL

pxxax

L

pxaxx

L

aa

aa


From (1) and (2)

Then re-arranging

2

21

1

12

11 1

p

xxa

p

xaxλ

aaaa-

1

2

2

1

1 xa

ax

p

p


Rearrange

Remember

axap

xp

1

2211

mxpxp 2211


Substitute

axap

xp

1

2211

mxpxp 2211


Solve x1* and x2*

and 1

*1 p

amx

2

*2

1

p

max

COMPUTING DEMAND Method 2

Suppose that the consumer has Cobb-Douglas preferences.

U x x x xa b( , )1 2 1 2


Suppose that the consumer has Cobb-Douglas preferences.

U x x x xa b( , )1 2 1 2

MUUx

ax xa b1

11

12

MUUx

bx xa b2

21 2

1


So the MRS is

MRSdxdx

U xU x

ax x

bx x

axbx

a b

a b

2

1

1

2

11

2

1 21

2

1

//

.


So the MRS is

At (x1*,x2*), MRS = -p1/p2 , i.e. the slope of the budget constraint.

MRSdxdx

U xU x

ax x

bx x

axbx

a b

a b

2

1

1

2

11

2

1 21

2

1

//

.


So the MRS is

At (x1*,x2*), MRS = -p1/p2 so

MRSdxdx

U xU x

ax x

bx x

axbx

a b

a b

2

1

1

2

11

2

1 21

2

1

//

.

ax

bx

pp

xbpap

x2

1

1

22

1

21

*

** *. (A)


(x1*,x2*) also exhausts the budget so

p x p x m1 1 2 2* * . (B)


So now we know that

xbpap

x21

21

* * (A)

p x p x m1 1 2 2* * . (B)


So now we know that

xbpap

x21

21

* * (A)

p x p x m1 1 2 2* * . (B)

Substitute


So now we know that

xbpap

x21

21

* * (A)

p x p x m1 1 2 2* * . (B)

p x pbpap

x m1 1 21

21

* * .

Substitute

and get

This simplifies to ….


xam

a b p11

*

( ).


2

*2 )( pba

bmx

Substituting for x1* in p x p x m1 1 2 2

* *

then gives

1

*1 )( pba

amx


So we have discovered that the mostpreferred affordable bundle for a consumerwith Cobb-Douglas preferences

U x x x xa b( , )1 2 1 2

is )(21

*2

*1 )(

,)(

),(pba

mb

pba

maxx

COMPUTING DEMAND Method 2: Cobb-Douglas

x1

x2

xam

a b p11

*

( )

x

bma b p

2

2

*

( )

U x x x xa b( , )1 2 1 2

Rational Constrained Choice

But what if x1* = 0 or x2* = 0?

If either x1* = 0 or x2* = 0 then the ordinary demand (x1*,x2*) is at a corner solution to the problem of maximizing utility subject to a budget constraint.

Examples of Corner Solutions: Perfect Substitutes

x1

x2

MRS = -1


x1

x2

MRS = -1

Slope = -p1/p2 with p1 > p2.


x1

x2

MRS = -1



x1

x2

2

*2 p

mx

x1 0*

MRS = -1 (This is the indifference curve)



x1

x2

1

*1 p

mx

x2 0*

MRS = -1

Slope = -p1/p2 with p1 < p2.

ANOTHER EXAMPLE


So when U(x1,x2) = x1 + x2, the mostpreferred affordable bundle is (x1*,x2*)where

0,),(

1

*2

*1 p

mxx

or

2

*2

*1 ,0),(

p

mxx

if p1 < p2

if p1 > p2.


x1

x2

MRS = -1

Slope = -p1/p2 with p1 = p2.

1p

m

2p

m

The budget constraint and the utility curve lie on each other


x1

x2

All the bundles in the constraint are equally the most preferred affordable when p1 = p2.

yp2

yp1

Examples of ‘Kinky’ Solutions: Perfect Complements

X1 (tonic)

X2 (gin) U(x1,x2) = min(ax1,x2)

x2 = ax1 (a = .5)


x1

x2

MRS = 0

U(x1,x2) = min(ax1,x2)

x2 = ax1


x1

x2

MRS = -

MRS = 0


x2 = ax1


x1

x2

MRS = -

MRS = 0

MRS is undefined


x2 = ax1


x1

x2U(x1,x2) = min(ax1,x2)

x2 = ax1


x1


x2 = ax1

Which is the mostpreferred affordable bundle?


x1


x2 = ax1

The most preferredaffordable bundle


x1


x2 = ax1

x1*

x2*


x1


x2 = ax1

x1*

x2*

and p1x1* + p2x2* = m


x1


x2 = ax1

x1*

x2*

(a) p1x1* + p2x2* = m(b) x2* = ax1*


(a) p1x1* + p2x2* = m; (b) x2* = ax1*


(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = m


(a) p1x1* + p2x2* = m; (b) x2* = ax1*.

Substitution from (b) for x2* in (a) gives p1x1* + p2ax1* = mwhich gives

21

*1 app

mx


(a) p1x1* + p2x2* = m; (b) x2* = ax1*.


21

*2

21

*1 ;

app

amx

app

mx


(a) p1x1* + p2x2* = m; (b) x2* = ax1*.


21

*2

21

*1 ;

app

amx

app

mx


x1


x2 = ax1

xm

p ap11 2

*

x

amp ap

2

1 2

*

Date post:	14-Jan-2016
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Rational Choice. CHOICE 1. Scarcity (income constraint) 2. Tastes (indifference map/utility...

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